From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class
This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well a...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2020-04-01
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| Series: | Concrete Operators |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/conop-2020-0007 |
| _version_ | 1819121962417389568 |
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| author | Massaneda Xavier Thomas Pascal J. |
| author_facet | Massaneda Xavier Thomas Pascal J. |
| author_sort | Massaneda Xavier |
| collection | DOAJ |
| description | This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class. |
| first_indexed | 2024-12-22T06:44:53Z |
| format | Article |
| id | doaj.art-314e534b48964126956398eb006bdb16 |
| institution | Directory Open Access Journal |
| issn | 2299-3282 |
| language | English |
| last_indexed | 2024-12-22T06:44:53Z |
| publishDate | 2020-04-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Concrete Operators |
| spelling | doaj.art-314e534b48964126956398eb006bdb162022-12-21T18:35:19ZengDe GruyterConcrete Operators2299-32822020-04-01719111510.1515/conop-2020-0007conop-2020-0007From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna classMassaneda Xavier0Thomas Pascal J.1Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007-Barcelona, CataloniaInstitut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, FranceThis survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.https://doi.org/10.1515/conop-2020-0007nevanlinna classinterpolating sequencescorona theoremsampling setssmirnov class30h1530h0530h80 |
| spellingShingle | Massaneda Xavier Thomas Pascal J. From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class Concrete Operators nevanlinna class interpolating sequences corona theorem sampling sets smirnov class 30h15 30h05 30h80 |
| title | From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class |
| title_full | From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class |
| title_fullStr | From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class |
| title_full_unstemmed | From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class |
| title_short | From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class |
| title_sort | from h∞ to 𝒩 pointwise properties and algebraic structure in the nevanlinna class |
| topic | nevanlinna class interpolating sequences corona theorem sampling sets smirnov class 30h15 30h05 30h80 |
| url | https://doi.org/10.1515/conop-2020-0007 |
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